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A166509
Decimal expansion of Sum_{p prime} (Kronecker(-1/p)/p).
4
1, 6, 5, 0, 1, 8, 6, 7, 4, 7, 0, 0, 0, 0, 6, 8, 1, 8, 9, 3, 6, 6, 8, 2, 8, 7, 8, 5, 1, 2, 4, 5, 6, 4, 2, 6, 2, 2, 0, 0, 2, 4, 6, 1, 9, 2, 4, 4, 9, 2, 2, 9, 5, 1, 8, 9, 1, 9, 7, 9, 4, 2, 1, 1, 5, 4, 7, 7, 7, 1, 5, 6, 7, 2, 8, 1, 1, 5, 8, 8, 9, 3, 7, 5, 1, 0, 0, 3, 6, 8, 9, 7, 0, 1, 9, 6, 6, 5, 4, 6, 0, 7, 5, 1, 3
OFFSET
0,2
COMMENTS
As a decimal number, this equals 0.5 - A086239. The Kronecker-Legendre-Jacobi symbol Kronecker(-1/p) equals -1 if p=3 (mod 4) and +1 for all other primes. The first ten digits have been computed using the numerical result from Finch for A086239.
Using results from Cohen's preprint, David Broadhurst computed 300 decimals in less than 0.2 seconds (cf. link to primenumbers group). He reports that the method can give 70 correct decimals using only two odd primes, p=3 and p=5. - M. F. Hasler, Oct 29 2009
Contrary to an earlier comment, this constant is not hard to evaluate: see the link for the first 10000 decimal digits. - David Broadhurst, Oct 30 2009
REFERENCES
Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
S. R. Finch, Meissel-Mertens Constants, Sect. 2.2 in Mathematical Constants. Cambridge, England: Cambridge University Press, 2003, p. 96.
LINKS
J. P. Benney and others, Is this a convergent series and if so what is its sum?, in primenumbers group, Oct 26 2009.
Julien Benney, David Broadhurst, Andrew J. Walker Is this a convergent series and if so what is its sum?, digest of 9 messages in primenumbers Yahoo group, Oct 26 - Oct 30, 2009.
David Broadhurst, 10000 digits [From David Broadhurst, Oct 30 2009]
David Broadhurst, 10000 digits [Cached copy]
Henri Cohen, High-precision computation of Hardy-Littlewood constants. [pdf copy, with permission]
Eric Weisstein's World of Mathematics, Prime Sums
EXAMPLE
0.1650186747... = 1/2 - 1/3 + 1/5 - 1/7 - 1/11 + 1/13 + 1/17 ... with negative sign iff p=3 (mod 4).
PROG
(PARI) p=1; sum( i=2, 10^6, kronecker(-1, p=nextprime(p+2))/p, .5) /* the first 10^6 terms give 0.16502... */
(PARI) /* The given parameters NP, N yield the correct result for default(realprecision, 300) in less than 0.5 seconds. Code based on David Broadhurst's script posted on primenumbers group. */
A166509(NP=70, N=115)={ local( P=vector(NP, k, (2-prime(k)%4)/prime(k)));
sum( s=1, N, moebius(s)/s*log( prod( k=2, #P, 1-P[k]^s,
if( s%2, if( s==1, Pi/4, sumalt( k=0, (-1)^k/(2*k+1)^s)), zeta(s)*(1-1/2^s)) /* end if s%2 */
)), sum(k=2, #P, P[k], 1/2.))} \\ M. F. Hasler, Oct 29 2009
CROSSREFS
Cf. A166510.
Sequence in context: A264807 A245973 A214128 * A200635 A197261 A010773
KEYWORD
cons,easy,nonn
AUTHOR
M. F. Hasler, Oct 28 2009
EXTENSIONS
More terms, computed using David Broadhurst's method, from M. F. Hasler, Oct 29 2009
STATUS
approved